Q.Find a vector in the direction of $\bar a=\hat i-2\hat j$ that has magnitude 7 units.

For a vector a and a real number (scalar) k, the vector ka has magnitude |k||a| and points in the same direction as a when k>0, the opposite direction when k<0, and is the zero vector when k=0; in particular the unit vector along a is a-hat = a/|a|, and a vector of any desired length k along a's direction is k times that unit vector. Because ka is always parallel to a, scalar multiplication gives the standard algebraic test for two non-zero vectors a and b to be collinear (parallel): they are collinear exactly when a = m b for some non-zero scalar m, equivalently w …